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Question

The maximum number of Boolean expressions that can be formed for the
function f(x, y, z) satisfying the relation f(~x,y,~z) = f(x,y,z) is ___________.

Answer 1

Boolean variables are three X,Y,Z

By Applying the condition f(~x,y,~z) = f(x,y,z)  
we are reducing the choice the function can have now the function f(~x,y,~z)  and  f(x,y,z) both need to have same value . as the function is boolean mean it can have value either 0 or 1 . 

if f(~x,y,~z) is 0 than f(x,y,z) need to have value 0 

if f(~x,y,~z) is 1 than f(x,y,z) need to have value 1 

we have 4 such pairs : f(0) f(5) , f(1) f(4) , f(2) f(7) , f(3) f(6) and each can have value 0 or 1 . 

so possible such boolean function are = 2^4 = 16 

Comments

Number of self-dual functions $= 2^{2^{n-1}}$

where $n=3$
Number of self-dual functions$ =2^{2^{2}}=2^{4}=16$

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@Lakshman Patel RJIT how function is self dual?

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@himgta 

To get a dual of any Boolean Expression, replace-

1. OR with AND i.e.$ +$ with $.$
2. AND with OR i.e. $.$ with $+$
3. $1$ with $0$
4. $0$ with $1$

Self-dual Functions$:-$When a function is equal to it's dual, it is called as a self-dual function.

https://math.stackexchange.com/questions/1602598/number-of-self-dual-functions-and-number-of-inputs-for-which-self-dual-function

https://www.gatevidyalay.com/self-dual-functions-dual-of-a-boolean-expression/

see this